Integral Problem: Solutions Say Divergent, Help Please!

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In summary, the integral \int 1/|x|\,dx is divergent and tends to +∞ or -∞, not -ln2 as you calculated. This is because you integrated across a singularity and your integral is incorrect.
  • #1
Jalo
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Homework Statement



yc8m.jpg



Homework Equations





The Attempt at a Solution



Ive solved it this way:
yc8m.jpg
= [ln |x|]1-2 = ln 1 - ln 2 = -ln2

However the solutions say the integral is divergent, therefore it should tend to +∞ or -∞

If someone could tell me what I've done wrong I'd appreciate!
Thanks
 
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  • #2
You did two things wrong.

1. Lesser problem: Your integral is incorrect. [itex]\int 1/|x|\,dx = \mathop{\mathrm{sgn}} x \, \ln |x|[/itex], not [itex]\ln |x|[/itex].

2. Huge problem: You integrated across a singularity.f(x)=1/|x| is positive everywhere. How could the integral of this function from -2 to 1 possibly be negative? That you obtained a negative result when the integrand is always positive and the integration interval is in the positive direction should have been a big warning sign indicating that you did something wrong.
 

FAQ: Integral Problem: Solutions Say Divergent, Help Please!

What is an integral problem?

An integral problem is a mathematical problem that involves finding the area under a curve or the antiderivative of a function. It is a fundamental concept in calculus and is used to solve a variety of real-world problems.

What does it mean when the solutions say "divergent"?

When the solutions of an integral problem say "divergent," it means that the solution is infinite or does not exist. This can happen when the function is not continuous or if the limits of integration are infinite.

How do I know if I have solved an integral problem correctly?

You can verify your solution by differentiating it and seeing if it matches the original function. Additionally, you can use computer software or a graphing calculator to graph the function and check if the area under the curve matches your solution.

What should I do if I am struggling with solving an integral problem?

If you are struggling with solving an integral problem, it is important to review the fundamental concepts of calculus and practice with simpler problems. You can also seek help from a tutor or teacher for additional guidance and clarification.

Are there any tips or tricks for solving integral problems?

One tip for solving integral problems is to first identify the type of integral (e.g. definite or indefinite) and then choose the appropriate method (e.g. substitution, integration by parts). It is also helpful to break the problem into smaller parts and use algebraic manipulation to simplify the integrand.

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